Schismatic Temperament - History of Schismatic Temperaments

History of Schismatic Temperaments

Historically significant is the 1/8-schisma tuning of Hermann von Helmholtz and Norwegian composer Eivind Groven. Helmholtz had a special Physharmonica (a harmonium by Schiedmayer) with 24 tones to the octave. Groven built an organ internally equipped with 36 tones to the octave which had the ability to adjust its tuning automatically during performances; the performer plays a familiar 12-key (per octave) keyboard and in most cases the mechanism will choose from among the three tunings for each key so that the chords played sound virtually in just intonation. Abstractly, 1/8-schisma tuning may be considered the analog, among schismatic tunings, of 1/4-comma meantone among meantone tunings, as it also has pure interval ratios of 2:1 and 5:4, though with much more accurate interval ratios of 3:2 and 6:5 (less than a quarter of a cent off from just intonation) than its meantone counterpart. A 1/9-schisma tuning has also been proposed by Eduardo Sabat-Garibaldi, who together with his students uses a 53-tone to the octave guitar with this tuning.

Mark Lindley and Ronald Turner-Smith argue that schismatic tuning was briefly in use during the late medieval period. This was not temperament but merely 12-tone Pythagorean tuning, though typically tuned from G♭ to B in ascending just fifths and descending just fourths, instead of the prevalent A♭ to C♯ or E♭ to G♯ schemes. This allowed concordant diminished fourths D-G♭, A-D♭, E-A♭, and B-E♭ and augmented seconds G♭-A, D♭-E, and A♭-B to be used in place of the discordant Pythagorean major thirds D-F♯, A-C♯, E-G♯, and B-D♯ and minor thirds F♯-A, C♯-E, and G♯-B, respectively. Justly tuned fifths and fourths generate a reasonable schismatic tuning and therefore schismatic is in some respects an easier way to introduce justly tuned thirds into a Pythagorean harmonic fabric than meantone. However, the result suffers from the same difficulties as just intonation – for example, the wolf B-G♭ here arises all too easily when availing oneself of the concordant schismatic substitutions just outlined – so it is not surprising that meantone temperament became the dominant tuning system by the early Renaissance. Helmholtz's and Groven's systems get around some, but not all, of these difficulties by including multiple tunings for each key on the keyboard, so that a particular note can be tuned as G♭ in some contexts and F♯ in others, for example.